arXiv subjects

Luis Arenas-Carmona

Publications and source records attributed to Luis Arenas-Carmona.

At least 19 recordsLinked to original sources

arXiv2026

On some representations of Metacyclic groups whose integral forms can be computed from a single residual representation

In a previous work we computed the number of integral forms, over its field of definition, of an irreducible representation of a dihedral group. Here we apply the theory of Bruhat-Tits buildings to give similar formulas for a wider family of metacyclic groups that have representations of arbitrarily large dimension. Occasionally, these formulas can be extended to groups containing a subgroup in that family.

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arXiv2026

Computing quaternionic representations via twisted forms of Bruhat-Tits trees

This work is devoted to the study of representations of finite subgroups of the group of units of quaternion division algebras over a global or local field arising from the inclusion via extension of scalars splitting the algebra. Following a question by Serre, we study the set $\mathrm{IF}$ of conjugacy classes of integral representations that are conjugates of the given representation over the field. The set $\mathrm{IF}$ is often called the set of integral forms in the literature. In previous works we have seen that, for a given representation, the set $\mathrm{IF}$ can be indexed by the vertex set of a suitable subgraph of the Bruhat-Tits tree for the special linear group. In this work, we describe a construction that allows the simultaneous study of the set $\mathrm{IF}$ over different splitting fields. For this, we devise and use a theory of twisted Galois form of Bruhat-Tits trees. With this tool, we explicitly compute, in most cases, the cardinality of $\mathrm{IF}$ for the representation of the classical quaternion group of order $8$ studied by Serre, Feit and others, as much as for other similar groups.

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arXiv2025

Two characterizations of Bass orders via branches

It has been known for some time that the orders in the four dimensional matrix algebra over a local field that can be written as a finite intersection of maximal orders are precisely those whose Gorenstein closure is Eichler. In this paper, a similar characterization is given for orders whose Gorenstein closure is a Bass order. A second characterization, this time for the Bass orders themselves, is given in terms of their branches, i.e., maximal subgraphs of the Bruhat-Tits tree whose vertices are orders containing them.

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arXiv2025

Two dimensional integral representations via branches of the Bruhat-Tits tree

We apply the theory of branches in Bruhat-Tits trees, developed in previous works by the second author and others, to the study of two dimensional representations of finite groups over the ring of integers of a number field. We provide a general strategy to perform these computations, and we give explicit formulas for some particular families.

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arXiv2025

Eichler orders, quotient graphs and random walks

We study the extent to which the quotient of the Bruhat-Tits tree at one place $Q$, associated to a genus of orders of maximal rank, can be computed from the analogous quotient at a different place $P$. We show that this computation can be carried out, except for a small set of vertices depending on $P$, but not on $Q$. We give some geometrical conditions on the quotient at $P$ that ensure that this exceptional set is empty. This generalizes the formulas from a previous work that allow the computation of the quotient graph at all places, for the genus of maximal orders over the projective line. The methods presented here yield similar results for other genera or other curves.

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arXiv2024

A Continued Fractions Theory for the completion of the Puiseux field

In this work, we study a continued fractions theory for the topological completion of the field of Puiseux series. As usual, we prove that any element in the completion can be developed as a unique continued fractions, whose coefficients are polynomials in roots of the variable, and that this approximation is the best ''rational'' Diophantine approximation of such element. Then, we interpret the preceding result in terms of the action of a suitable arithmetic subgroup of the special linear group on the Berkovich space defined over the said completion. We also explore the connections between points of type IV of the Berkovich space in terms of some ''non-convergent'' or ''undefined'' continued fractions, in a sense that we make precise in the text.

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arXiv2024

Diophantine Approximation in local function fields via Bruhat-Tit trees

We use the theory of arithmetic quotients of the Bruhat-Tits tree developed by Serre and others to obtain Dirichlet-style theorems for Diophantine approximation on global function fields. This approach allows us to find sharp values for the constants involved and, occasionally, explicit examples of badly approximable quadratic irrationals. Additionally, we can use this method to easily compute the measure of the set of elements that can be written as the limit of a sequence of ``better than expected'' approximants. All these results can be easily obtained via continued fractions when they are available, so that quotient graphs can be seen as a partial replacement of them when this fails to be the case.

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arXiv2021

Quotients of the Bruhat-Tits tree by arithmetic subgroups of special unitary groups

Let $K$ be the function field of a curve $C$ over a field $\mathbb{F}$ of either odd or zero characteristic. Following the work by Serre and Mason on $\mathrm{SL}_2$, we study the action of arithmetic subgroups of $\mathrm{SU}(3)$ on its corresponding Bruhat-Tits tree associated to a suitable completion of $K$. More precisely, we prove that the quotient graph "looks like a spider", in the sense that it is the union of a set of cuspidal rays (the "legs"), parametrized by an explicit Picard group, that are attached to a connected graph (the "body"). We use this description in order to describe these arithmetic subgroups as amalgamated products and study their homology. In the case where $\mathbb{F}$ is a finite field, we use a result by Bux, Köhl and Witzel in order to prove that the "body" is a finite graph, which allows us to get even more precise applications.

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arXiv2020

Branches in the Bruhat-Tits tree for local fields of even characteristic

We extend our previous computations for the relative positions of branches of quaternions to the case of local fields of even characteristic. This is a key step to understand the set of maximal orders containing a given suborder, which is useful, for instance, to compute relative spinor images, thus solving the selectivity problem. In our previous work, the results where given in terms of the quadratic defect. In the present context, we introduce and characterize an analogous concept for Artin-Schreier extensions. It is no longer useful to restrict our attention to orders generated by pure quaternions, as a separable quadratic extension contains no non-trivial element of null trace. In this work we state our result for an arbitrary pair of generators, for which we discuss a more general version of the Hilbert symbol in this context.

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arXiv2019

Simultaneous diagonalization of vector bundles

Grothendieck-Birkhoff Theorem states that every finite dimensional vector bundle over the projective line splits as the sum of one dimensional vector bundles. In this work we study simultaneous splittings of two dimensional vector bundles over a finite field using the theory of Eichler orders. In the sheaf-theoretical context, an Eichler order in a matrix algebra is the intersection of the sheaves of endomorphisms of two vector bundles, so characterizing split Eichler orders solves the problem of simultaneous splitting. We caracterize both the genera of Eichler orders containing only split Eichler orders and the genera containing only a finite number of non-split classes, in terms of a divisor-valued distance parametrizing the genera. This article shall not be published in its present form. These results will be included in the article "On genera containing non-split Eichler orders over function fields" [arXiv:1905.08244].

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arXiv2019

On the missing branches of the Bruhat-Tits tree

Let k be a local field and let A be the two-by-two matrix algebra over k. In our previous work we developed a theory that allows the computation of the set of maximal orders in A containing a given suborder. This set is given as a sub-tree of the Bruhat-Tits tree that is called the branch of the order. Branches have been used to study the global selectivity problem and also to compute local embedding numbers. They can usually be described in terms of two invariants. To compute these invariants explicitly, the strategy in our past work has been visualizing branches through the explicit representation of the Bruhat-Tits tree in terms of balls in k. This is easier for orders spanning a split commutative sub-algebra, i.e., an algebra isomorphic to (k x k). In the present work, we develop a theory of branches over field extension that can be used to extend our previous computations to orders spanning a field. We use the same idea to compute branches for orders generated by arbitrary pairs of non-nilpotent pure quaternions. In fact, the hypotheses on the generators are not essential.

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arXiv2019

On genera containing non-split Eichler orders over function fields

Grothendieck-Birkhoff Theorem states that every finite dimensional vector bundle over the projective line P1 splits as the sum of one dimensional vector bundles. This can be rephrased, in terms of orders, as stating that all maximal orders over the projective line in a matrix algebra split. In this work we study the extent to which this result can be generalized to Eichler orders when the base field F is finite. To be precise, we characterize both the genera of Eichler orders containing only split orders and the genera containing only a finite number of non-split conjugacy classes. The latter characterization is given for arbitrary projective curves over F. The method developed here also allows us to compute quotient graphs for some subgroups of $PGL_2(F[t])$ of arithmetical interest. This paper includes material from the unpublished work "Simultaneous diagonalization of vector bundles".

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arXiv2017

Branches on division algebras

We describe the set of maximal orders in a 2-by-2 matrix algebra over a non-commutative local division algebra B containing a given suborder, for certain important families of such suborders, including rings of integers of division subalgebras of B or most maximal semisimple commutative subalgebras.

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arXiv2016

On optimal embeddings and trees

We apply the theory of Bruhat-Tits trees to the study of optimal embeddings of two and three dimensional commutative orders into quaternion algebras. Specifically, we determine how many conjugacy classes of global Eichler orders in a quaternion algebra yield optimal representations of such orders. This completes the previous work by C. Maclachlan, who considered only Eichler orders of square free level and integral domains as sub-orders. The same technique is used in the second part of this work to compute local embedding numbers, extending previous results by J. Brzezinski.

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arXiv2014

On some branches of the Bruhat-Tits tree

We give an algorithm to explicitly compute the largest subtree, in the local Bruhat-Tits tree for PSL_2(k), whose vertices correspond to orders containing a given suborder H, in terms of a set of generators for H. The shape of this subtree is described, when it is finite, by a set of two invariants. We use our method to provide a full table for the invariants of an order generated by a pair of orthogonal pure quaternions. In a previous work, the first author showed that determining the shape of these local subtrees allows the computation of representation fields, a class field determining the set of isomorphism classes, in a genus O of orders of maximal rank in a fixed central simple algebra, containing an isomorphic copy of H. Some further applications are described here.

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arXiv2014

Roots of unity in definite quaternion orders

A commutative order in a quaternion algebra is called selective if it is embeds into some, but not all, the maximal orders in the algebra. It is known that a given quadratic order over a number field can be selective in at most one indefinite quaternion algebra. Here we prove that the order generated by a cubic root of unity is selective for any definite quaternion algebra over the rationals with a type number 3 or larger. The proof extends to a few other closely related orders.

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arXiv2014

Selectivity on division algebras

A commutative order in a central simple algebra over a number field is said to be selective if it embeds in some, but not all, the maximal orders in the algebra. We completely characterize selective orders in central division algebras, of dimension 9 or greater, in terms of the characterization of selective orders given by Chindburg and Friedman in the quaternionic case.

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arXiv2013

Spinor class field for generalized Eichler orders

We compute the spinor class field for a genus of orders, in a central simple algebra of higher dimension, that are intersections of two maximal orders. In particular, we compute the number of spinor genera in a genus of such orders, as the degree of an explicit extension of class fields.

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